Weighting Technique Implementation

Introduction

Problem

As it can be seen in 20_04_08_Ridge.html, there is a relation between the Probability the model assigns to a gene and the gene’s mean level of expression. This is a problem because we had previously discovered a bias in the SFARI scores related to mean level of expression (Preprocessing/Gandal/AllRegions/RMarkdowns/20_04_03_SFARI_genes.html), which means that this could be a confounding factor in our model and the reason why it seems to perform well, so we need to remove this bias to recover the true biological signal that is mixed with it and improve the quality of our model.

General idea

train model with equal weights for all samples

for l in loop:
  calculate bias
  correct weights to reduce bias
  retrain model
  
Return last model

Pseudocode

Parameters:

eta: Learning rate
T: Number of loops
D: Training data
H: Classification model
c: bias constraint
lambda: scaling factor for the weights
$w_i$ with $i=1,...,N$: Weights assigned to each sample

Pseudocode:

lambda = 0
w = [1, ..., 1]
c = std(meanExpr(D))

h  = train classifier H with lambda and w

for t in 1,,,T do
  bias = <h(x), c(x)>
  update lambda to lambda - eta*bias
  update weights_hat to exp(lambda*mean(c))
  update weights to w_hat/(1+w_hat) if y_i=1, 1/(1+w_hat) if y_i=0
  update h with new weights
  
Return h

Remove Bias

Demographic Parity as a measure of Bias

Using Demographic Parity as a measure of bias: A fair classifier h should make positive predictions each segment $G$ of the population at the same rate as in all of the population

This definition is for discrete segments of the population. Since our bias is found across all the population but in different measures depending on the mean level of expression of the gene, we have to adapt this definition to a continuous bias scenario

Demographic Parity for our problem: A fair classifier h should make positive predictions on genes with a certail mean level of expression at the same rate as in all of the genes in the dataset

Bias Metric

The original formula for the Demographic Parity bias is

$c(x,0) = 0 $ when the prediction is negative
$c(x,1) = \frac{g(x)}{Z_G}-1$ when the prediction is positive. Where $g(x)$ is the Kronecker delta to indicate if the sample belongs to the protected group and $Z_G$ is the proportion of the population that belongs to the group we want to protect against bias

Using this definitions in our problem:

$g(x):$ Since all our samples belong to the protected group, this would always be 1

$Z_G:$ Since all of our samples belong to the protected group, this would also always be 1

So our measure of bias $c(x,1) = \frac{1}{1}-1 = 0$ for all samples. This doesn’t work, so we need to adapt it to our continous case

Adaptation of the bias metric

We can use $c(x,1) = std(meanExpr(x))$ as the constraint function, this way, when we calculate the bias of the dataset:

$h(x)\cdot c(x)$ will only be zero if the positive samples are balanced around the mean expression, and the sign of the bias will indicate the direction of the bias

Calculating the Weights

Notes:

This model is only going to be used to obtain the optimal weights for the observations. The weights obtained in this model are going to be the ones used to train the final model afterwards
In the original model we would use oversampling as part of the training function, but here, since we are adding weights to the samples, the oversampling will have to be performed separately (otherwise the weights don’t match the number of observations), so it was moved inside the create_train_test_sets function

### DEFINE FUNCTIONS

create_train_test_sets = function(p, seed){
  
  # Get SFARI Score of all the samples so our train and test sets are balanced for each score
  sample_scores = dataset %>% mutate(ID = rownames(.)) %>% dplyr::select(ID) %>% 
                  left_join(biased_predictions %>% dplyr::select(ID, gene.score), by = 'ID') %>% 
                  mutate(gene.score = ifelse(is.na(gene.score), 'None', gene.score))

  set.seed(seed)
  train_idx = createDataPartition(sample_scores$gene.score, p = p, list = FALSE)
  train_set = dataset[train_idx,]
  test_set = dataset[-train_idx,]
  
  # Modify SFARI label in train set, save gene IDS (bc we lose them with SMOTE) and perform oversampling using SMOTE
  train_set = train_set %>% mutate(SFARI = ifelse(SFARI == TRUE, 'SFARI', 'not_SFARI') %>% as.factor,
                                   ID = rownames(.) %>% as.factor) %>% SMOTE(form = SFARI ~ . - ID)
  train_set_IDs = train_set %>% pull(ID)
  
  return(list('train_set' = train_set %>% dplyr::select(-ID), 'test_set' = test_set, 
              'train_set_IDs' = train_set_IDs))
}

run_weights_model = function(p, seed, Loops){
  
  # CREATE TRAIN AND TEST SETS
  train_test_sets = create_train_test_sets(p, seed)
  train_set = train_test_sets[['train_set']]
  test_set = train_test_sets[['test_set']]
  train_set_IDs = train_test_sets[['train_set_IDs']]
  
  
  # SET INITIAL PARAMETERS
  
  # General parameters
  set.seed(seed)
  lambda_seq = 10^seq(1, -4, by = -.1)
  k_fold = 10
  cv_repeats = 5
  trControl = trainControl(method = 'repeatedcv', number = k_fold, repeats = cv_repeats, verboseIter = FALSE, 
                           classProbs = TRUE, savePredictions = 'final', summaryFunction = twoClassSummary)
  # Bias correction parameters
  eta = 0.5
  lambda = 0
  w = rep(1, nrow(train_set))
  
  
  # TRAIN MODEL
  h = train(SFARI ~., data = train_set, method = 'glmnet', trControl = trControl, metric = 'ROC',
            tuneGrid = expand.grid(alpha = 0, lambda = lambda_seq))
  
  
  # CORRECT BIAS
  
  # Mean Expression info
  mean_expr = data.frame('ID' = train_set_IDs) %>% 
              left_join(data.frame('ID' = rownames(datExpr), 'meanExpr' = rowMeans(datExpr)), by = 'ID') %>%
              mutate('meanExpr_std' = (meanExpr-mean(meanExpr))/sd(meanExpr))
  
  # Track behaviour of plot
  bias_vec = c()
  acc_vec = c()
  
  for(l in 1:Loops){
    
    # Calculate bias for positive predicted samples
    bias = mean(mean_expr$meanExpr_std[predict(h, train_set)=='SFARI'])
    
    # Update weights
    lambda = lambda - eta*bias
    w_hat = exp(lambda*mean_expr$meanExpr_std)
    w = 1/(1+w_hat)
    w[train_set$SFARI=='SFARI'] = w[train_set$SFARI=='SFARI']*w_hat[train_set$SFARI=='SFARI']
    
    # Update tracking vars
    bias_vec = c(bias_vec, bias)
    acc_vec = c(acc_vec, mean(predict(h, train_set) == train_set$SFARI))
    
    # Update h
    h = train(SFARI ~., data = train_set, method = 'glmnet', weights = w, trControl = trControl, 
              metric = 'ROC', tuneGrid = expand.grid(alpha = 0, lambda = lambda_seq))
  }

  
  return(list('lambda' = lambda, 'bias_vec' = bias_vec, 'acc_vec' = acc_vec))
}


### RUN MODEL TO FIND OPTIMAL WEIGHTS

# Parameters
p = 0.75
seed = 123
Loops = 50

# Run model
model_output = run_weights_model(p, seed, Loops)

# Extract metrics
lambda = model_output[['lambda']]
bias_vec = model_output[['bias_vec']]
acc_vec = model_output[['acc_vec']]


rm(p, seeds, Loops, run_weights_model)

The optimal value of $\lambda$ is -1.7419319

The bias decreases until it oscilates around zero and the accuracy is not affected much

plot_info = data.frame('iter' = 1:length(bias_vec), 'bias' = bias_vec, 'accuracy' = acc_vec) %>% 
            melt(id.vars = 'iter')

plot_info %>% ggplot(aes(x=iter, y=value, color = variable)) + geom_line() + xlab('Iteration') + theme_minimal()

Since the bias increases the probability of being classified as 1 for genes with higher levels of expression, as the level of expression of a gene increases, the algorithm:

Increases the weight of genes with a negative label
Decreases the weight of genes with a positive label

mean_expr = data.frame('ID' = rownames(datExpr), 'meanExpr' = rowMeans(datExpr)) %>%
            left_join(predictions, by = 'ID') %>% filter(n>0) %>%
            mutate('meanExpr_std' = (meanExpr-mean(meanExpr))/sd(meanExpr))

w_hat = exp(lambda*mean_expr$meanExpr_std) # inverso a mean expr
w0 = 1/(1+w_hat) # prop a mean expr
w = 1/(1+w_hat)
w[mean_expr$SFARI %>% as.logical] = w[mean_expr$SFARI %>% as.logical]*w_hat[mean_expr$SFARI %>% as.logical] # inv mean expr Positives, prop Negatives
plot_data = data.frame('meanExpr' = mean_expr$meanExpr, 'w_hat' = w_hat, 'w0' = w0, 'w' = w, 
                       'SFARI' = mean_expr$SFARI, 'pred' = mean_expr$pred)

plot_data %>% ggplot(aes(meanExpr, w, color = SFARI)) + geom_point(alpha = 0.3) + 
              xlab('Mean Expression') + ylab('Weight') +
              ggtitle('Weights of the final model') + ylim(c(0,1)) + theme_minimal()

rm(mean_expr, w_hat, w0, w)

Running the final model

run_final_model = function(p, seed, lambda){
  
  # CREATE TRAIN AND TEST SETS
  train_test_sets = create_train_test_sets(p, seed)
  train_set = train_test_sets[['train_set']]
  test_set = train_test_sets[['test_set']]
  train_set_IDs = train_test_sets[['train_set_IDs']]
  
  
  # SET INITIAL PARAMETERS
  
  # General parameters
  set.seed(seed)
  lambda_seq = 10^seq(1, -4, by = -.1)
  k_fold = 10
  cv_repeats = 5
  trControl = trainControl(method = 'repeatedcv', number = k_fold, repeats = cv_repeats, verboseIter = FALSE, 
                           classProbs = TRUE, savePredictions = 'final', summaryFunction = twoClassSummary)
  
  # Bias correcting parameters
  mean_expr = data.frame('ID' = train_set_IDs) %>% 
              left_join(data.frame('ID' = rownames(datExpr), 'meanExpr' = rowMeans(datExpr)), by = 'ID') %>%
              mutate('meanExpr_std' = (meanExpr-mean(meanExpr))/sd(meanExpr))
  w_hat = exp(lambda*mean_expr$meanExpr_std)
  w = 1/(1+w_hat)
  w[train_set$SFARI=='SFARI'] = w[train_set$SFARI=='SFARI']*w_hat[train_set$SFARI=='SFARI']
  
  
  # TRAIN MODEL
  fit = train(SFARI ~., data = train_set, method = 'glmnet', weights = w, trControl = trControl, 
              metric = 'ROC', tuneGrid = expand.grid(alpha = 0, lambda = lambda_seq))
  
  
  # PREDICT TEST SET LABELS AND CREATE PERFORMANCE METRICS
  
  # Predict labels in test set
  predictions = fit %>% predict(test_set, type = 'prob')
  preds = data.frame('ID' = rownames(test_set), 'corrected_prob' = predictions$SFARI) %>% 
          mutate(corrected_pred = corrected_prob > 0.5)
  

  # Measure performance of the model
  acc = mean(test_set$SFARI==preds$corrected_pred)
  prec = Precision(test_set$SFARI %>% as.numeric, preds$corrected_pred %>% as.numeric, positive = '1')
  rec = Recall(test_set$SFARI %>% as.numeric, preds$corrected_pred %>% as.numeric, positive = '1')
  F1 = F1_Score(test_set$SFARI %>% as.numeric, preds$corrected_pred %>% as.numeric, positive = '1')
  pred_ROCR = prediction(preds$corrected_prob, test_set$SFARI)
  AUC = performance(pred_ROCR, measure='auc')@y.values[[1]]
  
  # Extract coefficients from features
  coefs = coef(fit$finalModel, fit$bestTune$lambda) %>% as.vector
  
  
  return(list('acc' = acc, 'prec' = prec, 'rec' = rec, 'F1' = F1, 'AUC' = AUC, 'preds' = preds, 'coefs'= coefs))
}


### RUN MODEL

# Parameters
p = 0.75
n_iter = 25
seeds = 123:(123+n_iter-1)

# So the input is the same as in 10_classification_model.html
original_dataset = dataset %>% mutate(ID = rownames(.)) %>% 
                   left_join(biased_predictions %>% dplyr::select(ID, gene.score))

# Store outputs
acc = c()
prec = c()
rec = c()
F1 = c()
AUC = c()
predictions = data.frame('ID' = rownames(dataset), 'SFARI' = dataset$SFARI, 'corrected_prob' = 0, 
                         'corrected_pred' = 0, 'n' = 0)
coefs = data.frame('var' = c('Intercept', colnames(dataset[,-ncol(dataset)])), 'coef' = 0)

for(seed in seeds){
  
  # Run model
  model_output = run_final_model(p, seed, lambda)
  
  # Update outputs
  acc = c(acc, model_output[['acc']])
  prec = c(prec, model_output[['prec']])
  rec = c(rec, model_output[['rec']])
  F1 = c(F1, model_output[['F1']])
  AUC = c(AUC, model_output[['AUC']])
  preds = model_output[['preds']]
  coefs$coef = coefs$coef + model_output[['coefs']]
  update_preds = preds %>% dplyr::select(-ID) %>% mutate(n=1)
  predictions[predictions$ID %in% preds$ID, c('corrected_prob','corrected_pred','n')] = 
    predictions[predictions$ID %in% preds$ID, c('corrected_prob','corrected_pred','n')] +
     update_preds
}

coefs = coefs %>% mutate(coef = coef/n_iter)
predictions = predictions %>% mutate(corrected_prob = corrected_prob/n, corrected_pred_count = corrected_pred, 
                                     corrected_pred = corrected_prob>0.5) %>% 
              left_join(biased_predictions %>% dplyr::select(ID, prob, pred), by = 'ID')


rm(p, seeds, update_preds, create_train_test_sets, run_final_model)

test_set = predictions %>% filter(n>0) %>% 
           left_join(dataset %>% mutate(ID = rownames(.)) %>% dplyr::select(ID, GS, MTcor), by = 'ID')
rownames(test_set) = predictions$ID[predictions$n>0]

Results

The relation between the model probability and the mean level of expression of the genes is not completely gone, there seems to be a negative relation for the genes with the lowest levels of expression

Even though the trend line is not as flat as with the first method, we are not fixing this directly as we were doing before, this is now just a consequence of the corrections we did inside of the model, so it makes sense for it to be less exact than before

# Plot results
plot_data = data.frame('ID'=rownames(datExpr), 'meanExpr'=rowMeans(datExpr)) %>% 
            right_join(test_set, by='ID')

plot_data %>% ggplot(aes(meanExpr, corrected_prob)) + geom_point(alpha=0.2, color='#0099cc') +
              geom_smooth(method='gam', color='gray', alpha=0.2) + 
              xlab('Mean Expression') + ylab('Corrected Probability') +
              theme_minimal() + ggtitle('Mean expression vs Model Probability corrected using adjusted Weights')

Performance metrics

Confusion matrix

conf_mat = test_set %>% apply_labels(SFARI = 'Actual Labels', 
                                     corrected_prob = 'Assigned Probability', 
                                     corrected_pred = 'Label Prediction')

cro(conf_mat$SFARI, list(conf_mat$corrected_pred, total()))

	Label Prediction		#Total
	FALSE	TRUE
Actual Labels
FALSE	12851	2351	15202
TRUE	559	221	780
#Total cases	13410	2572	15982

rm(conf_mat)

Accuracy: Mean = 0.8138 SD = 0.0242

Precision: Mean = 0.0874 SD = 0.011

Recall: Mean = 0.296 SD = 0.0541

F1 score: Mean = 0.1343 SD = 0.0166

ROC Curve: Mean = 0.6106 SD = 0.0196

pred_ROCR = prediction(test_set$corrected_prob, test_set$SFARI)

roc_ROCR = performance(pred_ROCR, measure='tpr', x.measure='fpr')
auc = performance(pred_ROCR, measure='auc')@y.values[[1]]

plot(roc_ROCR, main=paste0('ROC curve (AUC=',round(auc,2),')'), col='#009999')
abline(a=0, b=1, col='#666666')

Lift Curve

lift_ROCR = performance(pred_ROCR, measure='lift', x.measure='rpp')
plot(lift_ROCR, main='Lift curve', col='#86b300')

rm(pred_ROCR, roc_ROCR, AUC, lift_ROCR, acc, acc_vec, auc, bias_vec, F1, prec, rec)

Coefficients

MTcor and aGS have a very small coefficient and absGGS has the largest negative coefficient of all

gene_corr_info = dataset %>% mutate('ID' = rownames(dataset)) %>% dplyr::select(ID, MTcor, SFARI) %>% 
                 left_join(assigned_module, by ='ID') %>% mutate(Module = gsub('#','',Module))

coef_info = coefs %>% mutate('feature' = gsub('MM.','',var)) %>% 
            left_join(gene_corr_info, by = c('feature' = 'Module')) %>% 
            dplyr::select(feature, coef, MTcor, SFARI) %>% group_by(feature, coef, MTcor) %>% 
            summarise('SFARI_perc' = mean(SFARI)) %>% arrange(desc(coef))

coef_info %>% dplyr::select(feature, coef) %>% filter(feature %in% c('Intercept','GS','absGS','MTcor')) %>%
              dplyr::rename('Feature' = feature, 'Coefficient' = coef) %>% 
              kable(align = 'cc', caption = 'Regression Coefficients') %>% kable_styling(full_width = F)

Regression Coefficients
Feature	Coefficient
GS	0.1132241
MTcor	0.0026661
Intercept	-0.2344033
absGS	-0.6040255

There is still a positive relation between the coefficient assigned to the membership of each module and the enrichment (using ORA) in SFARI genes that are assigned to that module

load('./../Data/ORA.RData')

enrichment_SFARI_info = data.frame('Module'=as.character(), 'SFARI_enrichment'=as.numeric())
for(m in names(enrichment_SFARI)){
  m_info = enrichment_SFARI[[m]]
  enrichment = 1-ifelse('SFARI' %in% m_info$ID, m_info$pvalue[m_info$ID=='SFARI'],1)
  enrichment_SFARI_info = enrichment_SFARI_info %>% 
                          add_row(Module = gsub('#','',m), SFARI_enrichment = enrichment)
}

plot_data = coef_info %>% dplyr::rename('Module' = feature) %>% 
            left_join(enrichment_SFARI_info, by = 'Module') %>% filter(!is.na(MTcor))

ggplotly(plot_data %>% ggplot(aes(coef, SFARI_enrichment)) + 
         geom_smooth(method = 'lm', color = 'gray', alpha = 0.1) + 
         geom_point(aes(id = Module), color = paste0('#',plot_data$Module), alpha=0.7) + 
         theme_minimal() + xlab('Coefficient') + 
         ylab('SFARI Genes Enrichment'))

rm(enrichment_old_SFARI, enrichment_DGN, enrichment_DO, enrichment_GO, enrichment_KEGG, enrichment_Reactome, m,
   m_info, enrichment)

For the first time, there seems to be a possitive relation between the coefficients of the model and the correlation of the module and the diagnosis

ggplotly(coef_info %>% dplyr::rename('Module' = feature) %>% filter(!is.na(MTcor)) %>%
         ggplot(aes(coef, MTcor)) +  geom_smooth(method = 'lm', color = 'gray', alpha = 0.1) + 
         geom_point(aes(id = Module), color=paste0('#',coef_info$feature[!is.na(coef_info$MTcor)]), alpha=.7) + 
         theme_minimal() + xlab('Coefficient') + ylab('Module-Diagnosis correlation'))

Analyse Results

Probability distribution by SFARI Label

SFARI genes have a higher Probability distribution than the rest, but the overlap is larger than before

plot_data = test_set %>% dplyr::select(corrected_prob, SFARI)

ggplotly(plot_data %>% ggplot(aes(corrected_prob, fill=SFARI, color=SFARI)) + geom_density(alpha=0.3) + 
         geom_vline(xintercept = mean(plot_data$corrected_prob[plot_data$SFARI]), color = '#00C0C2', 
                    linetype='dashed') +
         geom_vline(xintercept = mean(plot_data$corrected_prob[!plot_data$SFARI]), color = '#FF7371', 
                    linetype='dashed') +
        xlab('Score') + ggtitle('Model Probability distribution by SFARI Label') + theme_minimal())

Probability distribution by SFARI Gene Scores

The relation between probability and SFARI Gene Scores weakened but it’s still there

plot_data = test_set %>% mutate(ID=rownames(test_set)) %>% dplyr::select(ID, corrected_prob) %>%
            left_join(original_dataset, by='ID') %>% dplyr::select(ID, corrected_prob, gene.score) %>% 
            apply_labels(gene.score='SFARI Gene score')

cro(plot_data$gene.score)

	#Total
SFARI Gene score
1	144
2	205
3	430
Neuronal	927
Others	14261
#Total cases	15967

mean_vals = plot_data %>% group_by(gene.score) %>% summarise(mean_prob = mean(corrected_prob))

comparisons = list(c('1','2'), c('2','3'), c('3','Neuronal'), c('Neuronal','Others'),
                   c('1','3'), c('3','Others'), c('2','Neuronal'),
                   c('1','Neuronal'), c('2','Others'), c('1','Others'))
increase = 0.07
base = 0.75
pos_y_comparisons = c(rep(base, 4), rep(base + increase, 2), base + 2:5*increase)

plot_data %>% filter(!is.na(gene.score)) %>% ggplot(aes(gene.score, corrected_prob, fill=gene.score)) + 
              geom_boxplot(outlier.colour='#cccccc', outlier.shape='o', outlier.size=3) +
              stat_compare_means(comparisons = comparisons, label = 'p.signif', method = 't.test', 
                                 method.args = list(var.equal = FALSE), label.y = pos_y_comparisons, 
                                 tip.length = .02) +
              scale_fill_manual(values=SFARI_colour_hue(r=c(1:3,8,7))) + 
              ggtitle('Distribution of probabilities by SFARI score') +
              xlab('SFARI score') + ylab('Probability') + theme_minimal() + theme(legend.position = 'none')

rm(mean_vals, increase, base, pos_y_comparisons)

Genes with the highest Probabilities

The concentration of SFARI genes decrease from 1:3 to 1:6
The genes with the highest probabilities are no longer SFARI Genes

test_set %>% dplyr::select(corrected_prob, SFARI) %>% mutate(ID = rownames(test_set)) %>% 
             arrange(desc(corrected_prob)) %>% top_n(50, wt=corrected_prob) %>%
             left_join(biased_predictions %>% dplyr::select(ID, gene.score, external_gene_id, MTcor, GS), 
                       by = 'ID') %>%
             dplyr::rename('GeneSymbol' = external_gene_id, 'Probability' = corrected_prob, 
                           'ModuleDiagnosis_corr' = MTcor, 'GeneSignificance' = GS) %>%
             mutate(ModuleDiagnosis_corr = round(ModuleDiagnosis_corr,4), Probability = round(Probability,4), 
                    GeneSignificance = round(GeneSignificance,4)) %>%
             left_join(assigned_module, by = 'ID') %>%
             dplyr::select(GeneSymbol, GeneSignificance, ModuleDiagnosis_corr, Module, Probability,
                           gene.score) %>%
             kable(caption = 'Genes with highest model probabilities from the test set') %>% 
             kable_styling(full_width = F)

Genes with highest model probabilities from the test set
GeneSymbol	GeneSignificance	ModuleDiagnosis_corr	Module	Probability	gene.score
CLMP	0.0360	-0.0127	#00BF7D	0.7478	Others
SNX25	0.0969	-0.0127	#00BF7D	0.7186	Others
HIVEP2	0.0188	-0.9003	#00BADE	0.7103	1
EGR1	0.2173	0.5341	#EF67EB	0.6962	Neuronal
CORO1A	0.2003	0.5341	#EF67EB	0.6836	Others
ARC	0.2509	0.5341	#EF67EB	0.6828	Neuronal
VPS13B	0.3442	0.6175	#44A0FF	0.6808	1
ITGAX	-0.0070	-0.0127	#00BF7D	0.6790	Others
HECTD2	0.1857	0.5341	#EF67EB	0.6776	Others
PLXNC1	-0.0093	0.5341	#EF67EB	0.6757	Others
PER2	0.1483	0.5341	#EF67EB	0.6756	2
TBC1D8	0.0962	0.7206	#FF66A9	0.6740	Others
SERPING1	0.2974	0.8479	#00C19A	0.6734	Others
PARVG	0.5663	0.7206	#FF66A9	0.6720	Others
PTPRO	0.1180	0.5341	#EF67EB	0.6713	Neuronal
IKZF1	0.5540	0.7206	#FF66A9	0.6711	Others
ITGAM	0.2995	0.7206	#FF66A9	0.6698	Others
EGR2	0.4858	0.5341	#EF67EB	0.6696	Neuronal
ARHGAP20	-0.0198	-0.9003	#00BADE	0.6687	Others
BDNF	0.3529	0.5341	#EF67EB	0.6686	Neuronal
NR4A3	0.4253	0.5341	#EF67EB	0.6671	Neuronal
SORCS3	0.0768	0.5341	#EF67EB	0.6656	Others
CSF1	0.6118	0.8479	#00C19A	0.6634	Others
SETD5	0.2826	0.6175	#44A0FF	0.6632	1
TRIM9	0.4827	0.5341	#EF67EB	0.6620	Others
AHI1	0.0158	-0.0127	#00BF7D	0.6614	Others
UNC79	-0.0418	-0.0127	#00BF7D	0.6608	2
ZBTB21	0.3410	0.5341	#EF67EB	0.6602	Others
ARPP21	0.4189	0.7651	#77AF00	0.6598	Others
SCAF4	0.0161	-0.5573	#00C0BF	0.6595	Others
NCOR1	0.4663	0.5341	#EF67EB	0.6590	2
EDC4	0.0701	-0.7233	#FF63B6	0.6578	Others
LRRC7	0.1828	-0.9003	#00BADE	0.6565	Others
BACH2	0.2821	0.5341	#EF67EB	0.6561	Others
AOAH	0.0888	0.7206	#FF66A9	0.6552	Others
SIK2	0.1608	-0.7233	#FF63B6	0.6549	Others
SAMSN1	0.4237	0.7206	#FF66A9	0.6546	Others
TP53BP1	0.4200	0.5341	#EF67EB	0.6543	Others
CSMD3	0.0773	0.3302	#F47B5B	0.6543	Others
CELF2	-0.0655	-0.9003	#00BADE	0.6536	Others
RAPH1	-0.0554	-0.3847	#00B7E7	0.6525	Others
PACSIN3	-0.0813	-0.9003	#00BADE	0.6524	Others
GADL1	0.6016	0.8479	#00C19A	0.6517	Others
PSME4	0.2117	0.5341	#EF67EB	0.6515	Others
RIMBP2	0.2592	0.5341	#EF67EB	0.6511	Others
TET3	0.4848	0.5341	#EF67EB	0.6506	Others
DOCK2	0.4748	0.7206	#FF66A9	0.6497	Others
MIDN	-0.0560	-0.5573	#00C0BF	0.6494	Others
CTTNBP2	0.1711	0.6175	#44A0FF	0.6492	2
CD72	0.2570	0.5341	#EF67EB	0.6490	Others

Negative samples distribution

The objective of this model is to identify candidate SFARI genes. For this, we are going to focus on the negative samples (the non-SFARI genes)

negative_set = test_set %>% filter(!SFARI)

negative_set_table = negative_set %>% apply_labels(corrected_prob = 'Assigned Probability', 
                                                   corrected_pred = 'Label Prediction')

cro(negative_set_table$corrected_pred)

	#Total
Label Prediction
FALSE	12851
TRUE	2351
#Total cases	15202

NA genes are predicted as ASD-related

negative_set %>% ggplot(aes(corrected_prob)) + geom_density(color='#F8766D', fill='#F8766D', alpha=0.5) +
                 geom_vline(xintercept=0.5, color='#333333', linetype='dotted') + xlab('Probability') +
                 ggtitle('Probability distribution of the Negative samples in the Test Set') + 
                 theme_minimal()

negative_set %>% dplyr::select(corrected_prob, SFARI) %>% mutate(ID = rownames(negative_set)) %>% 
                 arrange(desc(corrected_prob)) %>% top_n(50, wt=corrected_prob) %>%
                 left_join(biased_predictions %>% dplyr::select(ID, gene.score, external_gene_id, MTcor, GS), 
                           by = 'ID') %>%
                 dplyr::rename('GeneSymbol' = external_gene_id, 'Probability' = corrected_prob, 
                               'ModuleDiagnosis_corr' = MTcor, 'GeneSignificance' = GS) %>%
                 mutate(ModuleDiagnosis_corr = round(ModuleDiagnosis_corr,4), 
                        Probability = round(Probability,4), 
                        GeneSignificance = round(GeneSignificance,4)) %>%
                 left_join(assigned_module, by = 'ID') %>%
                 dplyr::select(GeneSymbol, GeneSignificance, ModuleDiagnosis_corr, Module, Probability,
                               gene.score) %>%
                 kable(caption = 'Genes with highest model probabilities from the Negative set') %>% 
                 kable_styling(full_width = F)

Genes with highest model probabilities from the Negative set
GeneSymbol	GeneSignificance	ModuleDiagnosis_corr	Module	Probability	gene.score
CLMP	0.0360	-0.0127	#00BF7D	0.7478	Others
SNX25	0.0969	-0.0127	#00BF7D	0.7186	Others
EGR1	0.2173	0.5341	#EF67EB	0.6962	Neuronal
CORO1A	0.2003	0.5341	#EF67EB	0.6836	Others
ARC	0.2509	0.5341	#EF67EB	0.6828	Neuronal
ITGAX	-0.0070	-0.0127	#00BF7D	0.6790	Others
HECTD2	0.1857	0.5341	#EF67EB	0.6776	Others
PLXNC1	-0.0093	0.5341	#EF67EB	0.6757	Others
TBC1D8	0.0962	0.7206	#FF66A9	0.6740	Others
SERPING1	0.2974	0.8479	#00C19A	0.6734	Others
PARVG	0.5663	0.7206	#FF66A9	0.6720	Others
PTPRO	0.1180	0.5341	#EF67EB	0.6713	Neuronal
IKZF1	0.5540	0.7206	#FF66A9	0.6711	Others
ITGAM	0.2995	0.7206	#FF66A9	0.6698	Others
EGR2	0.4858	0.5341	#EF67EB	0.6696	Neuronal
ARHGAP20	-0.0198	-0.9003	#00BADE	0.6687	Others
BDNF	0.3529	0.5341	#EF67EB	0.6686	Neuronal
NR4A3	0.4253	0.5341	#EF67EB	0.6671	Neuronal
SORCS3	0.0768	0.5341	#EF67EB	0.6656	Others
CSF1	0.6118	0.8479	#00C19A	0.6634	Others
TRIM9	0.4827	0.5341	#EF67EB	0.6620	Others
AHI1	0.0158	-0.0127	#00BF7D	0.6614	Others
ZBTB21	0.3410	0.5341	#EF67EB	0.6602	Others
ARPP21	0.4189	0.7651	#77AF00	0.6598	Others
SCAF4	0.0161	-0.5573	#00C0BF	0.6595	Others
EDC4	0.0701	-0.7233	#FF63B6	0.6578	Others
LRRC7	0.1828	-0.9003	#00BADE	0.6565	Others
BACH2	0.2821	0.5341	#EF67EB	0.6561	Others
AOAH	0.0888	0.7206	#FF66A9	0.6552	Others
SIK2	0.1608	-0.7233	#FF63B6	0.6549	Others
SAMSN1	0.4237	0.7206	#FF66A9	0.6546	Others
TP53BP1	0.4200	0.5341	#EF67EB	0.6543	Others
CSMD3	0.0773	0.3302	#F47B5B	0.6543	Others
CELF2	-0.0655	-0.9003	#00BADE	0.6536	Others
RAPH1	-0.0554	-0.3847	#00B7E7	0.6525	Others
PACSIN3	-0.0813	-0.9003	#00BADE	0.6524	Others
GADL1	0.6016	0.8479	#00C19A	0.6517	Others
PSME4	0.2117	0.5341	#EF67EB	0.6515	Others
RIMBP2	0.2592	0.5341	#EF67EB	0.6511	Others
TET3	0.4848	0.5341	#EF67EB	0.6506	Others
DOCK2	0.4748	0.7206	#FF66A9	0.6497	Others
MIDN	-0.0560	-0.5573	#00C0BF	0.6494	Others
CD72	0.2570	0.5341	#EF67EB	0.6490	Others
GIT2	-0.1432	-0.0127	#00BF7D	0.6480	Others
DNAJB5	0.5220	0.5341	#EF67EB	0.6469	Others
DUSP6	0.4091	0.5341	#EF67EB	0.6463	Others
ANKRD36	-0.0351	-0.0127	#00BF7D	0.6448	Others
NR4A1	0.4369	0.9612	#88AC00	0.6447	Others
CD37	0.4633	0.7206	#FF66A9	0.6442	Others
DDX27	0.2347	-0.0127	#00BF7D	0.6441	Others

Comparison with the original model’s probabilities:

The genes with the highest Probabilitys were affected the most as a group
Iin general genes with the lowest Probabilities got their score increased and the genes with the highest scores, decreased
The change in Probability by gene is much larger than with the Post Processing approach

negative_set %>% mutate(diff = abs(prob-corrected_prob)) %>% 
             ggplot(aes(prob, corrected_prob, color = diff)) + geom_point(alpha=0.2) + scale_color_viridis() + 
             geom_abline(slope=1, intercept=0, color='gray', linetype='dashed') + 
             geom_smooth(color='#666666', alpha=0.5, se=TRUE, size=0.5) + coord_fixed() +
             xlab('Original probability') + ylab('Corrected probability') + theme_minimal() + theme(legend.position = 'none')

negative_set_table = negative_set %>% apply_labels(corrected_prob = 'Corrected Probability', 
                                                   corrected_pred = 'Corrected Class Prediction',
                                                   pred = 'Original Class Prediction') %>%
                     filter(!is.na(pred))

cro(negative_set_table$pred, list(negative_set_table$corrected_pred, total()))

	Corrected Class Prediction		#Total
	FALSE	TRUE
Original Class Prediction
FALSE	11168	949	12117
TRUE	1671	1400	3071
#Total cases	12839	2349	15188

83% of the genes maintained their original predicted class

rm(negative_set_table)

Probability and Gene Significance

The relation is the opposite as before, the higher the Gene Significance, the higher the probability, with the lowest probabilities corresponding to under-expressed genes

*The transparent verison of the trend line is the original trend line

negative_set %>% ggplot(aes(corrected_prob, GS, color=MTcor)) + geom_point() + 
                 geom_smooth(method='gam', color='#666666') + ylab('Gene Significance') +
                 geom_line(stat='smooth', method='gam', color='#666666', alpha=0.5, size=1.2, aes(x=prob)) +
                 geom_hline(yintercept=mean(negative_set$GS), color='gray', linetype='dashed') +
                 scale_color_gradientn(colours=c('#F8766D','white','#00BFC4')) + xlab('Corrected Probability') +
                 ggtitle('Relation between the Model\'s Corrected Probability and Gene Significance') + 
                 theme_minimal()

Summarised version of Probability vs mean expression, plotting by module instead of by gene

The difference in the trend lines between this plot and the one above is that the one above takes all the points into consideration while this considers each module as an observation by itself, so the top one is strongly affected by big modules and the bottom one treats all modules the same

The transparent version of each point and trend lines are the original values and trends before the bias correction

plot_data = negative_set %>% mutate(ID = rownames(.)) %>% left_join(assigned_module, by = 'ID') %>%
            group_by(MTcor, Module) %>% summarise(mean = mean(prob), sd = sd(prob),
                                                  new_mean = mean(corrected_prob),
                                                  new_sd = sd(corrected_prob), n = n()) %>%
            mutate(MTcor_sign = ifelse(MTcor>0, 'Positive', 'Negative')) %>% 
            dplyr::select(Module, MTcor, MTcor_sign, mean, new_mean, sd, new_sd, n) %>% distinct()
colnames(plot_data)[1] = 'ID'

ggplotly(plot_data %>% ggplot(aes(MTcor, new_mean, size=n, color=MTcor_sign)) + geom_point(aes(id = ID)) + 
         geom_smooth(method='loess', color='gray', se=FALSE) + geom_smooth(method='lm', se=FALSE) + 
         geom_point(aes(y=mean), alpha=0.3) + 
         geom_line(stat='smooth', method='loess', color='gray', se=FALSE, alpha=0.3, size=1.2, aes(y=mean)) + 
         geom_line(stat='smooth', method='lm', se=FALSE, alpha=0.3, size=1.2, aes(y=mean)) + 
         xlab('Module-Diagnosis correlation') + ylab('Mean Corrected Probability by Module') + 
         theme_minimal() + theme(legend.position='none'))

Probability and mean level of expression

To check if correcting by gene also corrected by module: Yes, the bias seems to be removed completely, it may even be a bit negative now

mean_and_sd = data.frame(ID=rownames(datExpr), meanExpr=rowMeans(datExpr), sdExpr=apply(datExpr,1,sd))

plot_data = negative_set %>% mutate(ID=rownames(test_set)[!test_set$SFARI]) %>% 
            left_join(mean_and_sd, by='ID') %>% 
            left_join(assigned_module, by='ID')

plot_data2 = plot_data %>% group_by(Module) %>% summarise(meanExpr = mean(meanExpr), meanProb = mean(prob), 
                                                          new_meanProb = mean(corrected_prob), n=n())

ggplotly(plot_data2 %>% ggplot(aes(meanExpr, new_meanProb, size=n)) + 
         geom_point(color=plot_data2$Module) + geom_point(color=plot_data2$Module, alpha=0.3, aes(y=meanProb)) + 
         geom_smooth(method='loess', se=TRUE, color='gray', alpha=0.1, size=0.7) + 
         geom_line(stat='smooth', method='loess', se=TRUE, color='gray', alpha=0.4, size=1.2, aes(y=meanProb)) +
         xlab('Mean Expression') + ylab('Corrected Probability') +  
         ggtitle('Mean expression vs corrected Model Probability by Module') +
         theme_minimal() + theme(legend.position='none'))

rm(plot_data2, mean_and_sd)

Probability and LFC

For under-expressed genes, the relation between LFC and probability got inverted. The difference is quite big

For over-expressed genes, the trend didn’t change, it just got translated higher. Now in general, over-expressed genes have higher probabilities than over-expressed genes

plot_data = negative_set %>% mutate(ID=rownames(test_set)[!test_set$SFARI]) %>% 
            left_join(DE_info %>% data.frame %>% mutate(ID=rownames(.)), by='ID')

plot_data %>% ggplot(aes(log2FoldChange, corrected_prob)) + geom_point(alpha=0.1, color='#0099cc') + 
              geom_smooth(method='loess', color='gray', alpha=0.1) + 
              geom_line(stat='smooth', method='loess', color='gray', alpha=0.4, size=1.5, aes(y=prob)) +
              xlab('LFC') + ylab('Corrected Probability') +
              theme_minimal() + ggtitle('LFC vs model probability by gene')

The trends for DE genes don’t seem to have changed much, they were just translated, upwards for over-expressed genes, and downwards for under-expressed genes

The big difference we see in the plot above is just the connection between the patterns for under and over-expressed genes

The group that seem to be affected the least is the non-DE group of under-expressed genes

p1 = plot_data %>% filter(log2FoldChange<0) %>% mutate(DE = padj<0.05) %>% 
     ggplot(aes(log2FoldChange, corrected_prob, color=DE)) + geom_point(alpha=0.1) + 
     geom_smooth(method='loess', alpha=0.1) + xlab('') + ylab('Corrected Probability') + 
     ylim(c(min(plot_data$corrected_prob), max(plot_data$corrected_prob))) + 
     geom_line(stat='smooth', method='loess', alpha=0.4, size=1.5, aes(y=prob, color = DE)) +
     theme_minimal() + theme(legend.position = 'none', plot.margin=unit(c(1,-0.3,1,1), 'cm'))

p2 = plot_data %>% filter(log2FoldChange>=0) %>% mutate(DE = padj<0.05) %>% 
     ggplot(aes(log2FoldChange, corrected_prob, color=DE)) + geom_point(alpha=0.1) + 
     geom_smooth(method='loess', alpha=0.1) + xlab('') + ylab('Corrected Probability') + ylab('') +
     scale_y_continuous(position = 'right', 
                        limits = c(min(plot_data$corrected_prob), max(plot_data$corrected_prob))) +
     geom_line(stat='smooth', method = 'loess', alpha=0.4, size=1.5, aes(y = prob, color = DE)) +
     theme_minimal() + theme(plot.margin = unit(c(1,1,1,-0.3), 'cm'), axis.ticks.y = element_blank())

grid.arrange(p1, p2, nrow=1, top = 'LFC vs model probability by gene', bottom = 'LFC')

rm(p1, p2)

Probability and Module-Diagnosis correlation

The Probabilitys decreased for modules with negative correlation and increased for modules with positive correlation

module_score = negative_set %>% mutate(ID=rownames(test_set)[!test_set$SFARI]) %>%
               left_join(biased_predictions %>% dplyr::select(ID, gene.score), by='ID') %>%
               left_join(assigned_module, by = 'ID') %>%
               dplyr::select(ID, prob, corrected_prob, Module, MTcor) %>% 
               left_join(data.frame(MTcor=unique(dataset$MTcor)) %>% arrange(by=MTcor) %>% 
                         mutate(order=1:length(unique(dataset$MTcor))), by='MTcor')

ggplotly(module_score %>% ggplot(aes(MTcor, corrected_prob)) + 
         geom_point(color=module_score$Module, aes(id=ID, alpha=corrected_prob^4)) +
         geom_hline(yintercept=mean(module_score$corrected_prob), color='gray', linetype='dotted') + 
         geom_line(stat='smooth', method = 'loess', color='gray', alpha=0.5, size=1.5, aes(x=MTcor, y=prob)) +
         geom_smooth(color='gray', method = 'loess', se = FALSE, alpha=0.3) + theme_minimal() + 
         xlab('Module-Diagnosis correlation') + ylab('Corrected Probability'))

Conclusion

This bias correction makes bigger changes in the distribution of the probabilities than the post-processing one. Its main effect seems to be to reduce the importance of the under-expressed genes and increase the importance of over-expressed genes

In general the performance metrics decrease, but this isn’t necessarily wrong, since we knew part of the good performance of the model was because of the confounding factor related to mean level of expression, so it was expected for the performance of the model to decrease once we have removed this signal

Saving results

write.csv(test_set, file='./../Data/RM_weighting_bias_correction.csv', row.names = TRUE)

Session info

sessionInfo()

## R version 3.6.3 (2020-02-29)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 18.04.4 LTS
## 
## Matrix products: default
## BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
## 
## locale:
##  [1] LC_CTYPE=en_GB.UTF-8       LC_NUMERIC=C              
##  [3] LC_TIME=en_GB.UTF-8        LC_COLLATE=en_GB.UTF-8    
##  [5] LC_MONETARY=en_GB.UTF-8    LC_MESSAGES=en_GB.UTF-8   
##  [7] LC_PAPER=en_GB.UTF-8       LC_NAME=C                 
##  [9] LC_ADDRESS=C               LC_TELEPHONE=C            
## [11] LC_MEASUREMENT=en_GB.UTF-8 LC_IDENTIFICATION=C       
## 
## attached base packages:
## [1] grid      stats     graphics  grDevices utils     datasets  methods  
## [8] base     
## 
## other attached packages:
##  [1] expss_0.10.2       kableExtra_1.1.0   MLmetrics_1.1.1    car_3.0-7         
##  [5] carData_3.0-3      ROCR_1.0-7         gplots_3.0.3       DMwR_0.4.1        
##  [9] caret_6.0-86       lattice_0.20-41    mgcv_1.8-31        nlme_3.1-147      
## [13] reshape2_1.4.4     ggpubr_0.2.5       magrittr_1.5       RColorBrewer_1.1-2
## [17] gridExtra_2.3      viridis_0.5.1      viridisLite_0.3.0  plotly_4.9.2      
## [21] knitr_1.28         forcats_0.5.0      stringr_1.4.0      dplyr_1.0.0       
## [25] purrr_0.3.4        readr_1.3.1        tidyr_1.1.0        tibble_3.0.1      
## [29] ggplot2_3.3.2      tidyverse_1.3.0   
## 
## loaded via a namespace (and not attached):
##   [1] readxl_1.3.1                backports_1.1.8            
##   [3] Hmisc_4.4-0                 plyr_1.8.6                 
##   [5] lazyeval_0.2.2              splines_3.6.3              
##   [7] crosstalk_1.1.0.1           BiocParallel_1.18.1        
##   [9] GenomeInfoDb_1.20.0         digest_0.6.25              
##  [11] foreach_1.5.0               htmltools_0.4.0            
##  [13] gdata_2.18.0                fansi_0.4.1                
##  [15] memoise_1.1.0               checkmate_2.0.0            
##  [17] cluster_2.1.0               openxlsx_4.1.4             
##  [19] annotate_1.62.0             recipes_0.1.10             
##  [21] modelr_0.1.6                gower_0.2.1                
##  [23] matrixStats_0.56.0          xts_0.12-0                 
##  [25] jpeg_0.1-8.1                colorspace_1.4-1           
##  [27] blob_1.2.1                  rvest_0.3.5                
##  [29] haven_2.2.0                 xfun_0.12                  
##  [31] crayon_1.3.4                RCurl_1.98-1.2             
##  [33] jsonlite_1.7.0              genefilter_1.66.0          
##  [35] survival_3.1-12             zoo_1.8-8                  
##  [37] iterators_1.0.12            glue_1.4.1                 
##  [39] gtable_0.3.0                ipred_0.9-9                
##  [41] zlibbioc_1.30.0             XVector_0.24.0             
##  [43] webshot_0.5.2               DelayedArray_0.10.0        
##  [45] shape_1.4.4                 BiocGenerics_0.30.0        
##  [47] quantmod_0.4.17             abind_1.4-5                
##  [49] scales_1.1.1                DBI_1.1.0                  
##  [51] Rcpp_1.0.4.6                xtable_1.8-4               
##  [53] htmlTable_1.13.3            bit_1.1-15.2               
##  [55] foreign_0.8-76              Formula_1.2-3              
##  [57] stats4_3.6.3                lava_1.6.7                 
##  [59] prodlim_2019.11.13          glmnet_3.0-2               
##  [61] htmlwidgets_1.5.1           httr_1.4.1                 
##  [63] acepack_1.4.1               ellipsis_0.3.1             
##  [65] farver_2.0.3                XML_3.99-0.3               
##  [67] pkgconfig_2.0.3             nnet_7.3-14                
##  [69] dbplyr_1.4.2                locfit_1.5-9.4             
##  [71] labeling_0.3                AnnotationDbi_1.46.1       
##  [73] tidyselect_1.1.0            rlang_0.4.6                
##  [75] munsell_0.5.0               cellranger_1.1.0           
##  [77] tools_3.6.3                 cli_2.0.2                  
##  [79] RSQLite_2.2.0               generics_0.0.2             
##  [81] broom_0.5.5                 evaluate_0.14              
##  [83] yaml_2.2.1                  bit64_0.9-7                
##  [85] ModelMetrics_1.2.2.2        fs_1.4.0                   
##  [87] zip_2.0.4                   caTools_1.18.0             
##  [89] xml2_1.2.5                  compiler_3.6.3             
##  [91] rstudioapi_0.11             png_0.1-7                  
##  [93] curl_4.3                    ggsignif_0.6.0             
##  [95] reprex_0.3.0                geneplotter_1.62.0         
##  [97] stringi_1.4.6               highr_0.8                  
##  [99] Matrix_1.2-18               vctrs_0.3.1                
## [101] pillar_1.4.4                lifecycle_0.2.0            
## [103] data.table_1.12.8           bitops_1.0-6               
## [105] GenomicRanges_1.36.1        latticeExtra_0.6-29        
## [107] R6_2.4.1                    KernSmooth_2.23-17         
## [109] rio_0.5.16                  IRanges_2.18.3             
## [111] codetools_0.2-16            MASS_7.3-51.6              
## [113] gtools_3.8.2                assertthat_0.2.1           
## [115] SummarizedExperiment_1.14.1 DESeq2_1.24.0              
## [117] withr_2.2.0                 S4Vectors_0.22.1           
## [119] GenomeInfoDbData_1.2.1      parallel_3.6.3             
## [121] hms_0.5.3                   rpart_4.1-15               
## [123] timeDate_3043.102           class_7.3-17               
## [125] rmarkdown_2.1               TTR_0.23-6                 
## [127] pROC_1.16.2                 base64enc_0.1-3            
## [129] Biobase_2.44.0              lubridate_1.7.4

Bias Correction Weighting Technique